Constructing School Timetables using Simulated Annealing

17 Pages·2007·0.05 MB·English

Solving Timetables using Simulated Annealing Page 1
Constructing School Timetables using
Simulated Annealing:
Sequential and Parallel Algorithms
D. Abramson ּ†
† Division of Information Technology
C.S.I.R.O.
c/o Department of Communication and Electronic Engineering
Royal Melbourne Institute of Technology
P.O. Box 2476V
Melbourne 3001
Australia.
Management Science, Vol 37, No 1, Jan 1991, pp 98 - 113.
ABSTRACT:
This paper considers a solution to the school timetabling problem. The
timetabling problem involves scheduling a number of tuples, each consisting of class
of students, a teacher, a subject and a room, to a fixed number of time slots. A Monte
Carlo scheme called simulated annealing is used as an optimisation technique. The
paper introduces the timetabling problem, and then describes the simulated annealing
method. Annealing is then applied to the timetabling problem. A prototype
timetabling environment is described followed by some experimental results. A
parallel algorithm which can be implemented on a multiprocessor is presented. This
algorithm can provide a faster solution than the equivalent sequential algorithm. Some
further experimental results are given.
1 INTRODUCTION
This paper considers a solution to the school timetabling problem. The timetabling
problem involves scheduling a number of tuples, each consisting of class of students, a teacher,
a subject and a room, to a fixed number of time slots. A number of such tuples may be scheduled
in the same time slot providing no class, teacher or room appears more than once in the time slot.
This problem has been well studied in the past and some limited success has been
reported. An exhaustive search is impratical because there are too many alternatives. Modelling
the problem as an integer programming problem has not been particularly successful because
there are too many variables and constraints [2,3,4,6,7,8]. Heuristic searches have had
reasonable success, but it is hard to form a heuristic that performs as well as an experienced
human [1]. Monte Carlo techniques have been used to a limited extent [5,9]. In this paper we
examine the use of a Monte Carlo scheme called simulated annealing as an optimisation
technique. We introduce the timetabling problem, followed by the simulated annealing technique.
We then show how annealing can be applied to the timetabling problem. We describe a prototype
timetabling environment, and give some experimental results. The timetabling package which has
been developed is based on the theory given in this paper. It can be used by timetable planners in
preparing a usable schedule of classes and teachers. The data is entered as a set of requirements
and constraints, as illustrated in section 9. The package manipulates the positions of the various
classes until it has minimised the number of clashes in the timetable. The output is either a valid
schedule, or one which is as close as possible to a valid schedule. The various input constraints
are described in section 5.
Solving Timetables using Simulated Annealing Page 2
A parallel algorithm which can be implemented on a multiprocessor is presented. This
algorithm can provide a faster solution than the equivalent sequential algorithm. Some further
experimental results are given.
The timetabling problem is important not only because many schools and universities
produce timetables each year, but also because it is just one of many scheduling problems. An
effective solution to the timetabling problem could be applied to other constraint based
scheduling tasks. The parallelisation of the serial algorithm could be effective as a technique for
other simulated annealing algorithms.
2 TIMETABLING PROBLEM
The problem of creating a valid timetable involves scheduling classes, teachers and rooms
in such a way that no teacher, class or room is used more than once per period. For example, if a
class must meet twice a week, then it must be placed in two different periods to avoid a clash.
The timetable is to be distributed across a fixed number of periods per week. A class consists of
a number of students. We will assume that students have already been grouped into classes.
Initially we consider classes to be disjoint, that is, they have no students in common. In this
scheme, a correct timetable is one in which a class can be scheduled concurrently with any other
class. Later in the paper this restriction will be relaxed to cater for high schools and Universities
in which students can take many options. In each period a class is taught a subject. It is possible
for a subject to appear more than once in a period. A particular combination of a teacher, a
subject, a room and a class is called an element. An element may be required more than once per
week. The combination of an element and a frequency is called a requirement. Thus, the
timetabling problem can be phrased as scheduling a number of requirements such that a
requirement, teacher, class or room does not appear more than once per period.
The task of combining a class, teacher and room combination into a requirement is
handled as a separate operation, and is not the subject of this paper. They are formed with
knowledge of the requirements of the various students and teachers, as well as which rooms are
available. It is convenient to partition the problem in this way as it reduces the complexity of the
scheduling task into two smaller allocation problems. In many cases it is possible to form the
requirements without the need to use an optimisation scheme because the relationship between
classes of students, teachers and rooms is often fixed. Later in the paper a scheme is described in
which the rooms need not be allocated at this stage, but only a group of possible rooms each
with the same characteristics. Thus, it becomes only necessary to combine classes with their
teachers.
A sample timetable is shown in Figure 1. A CLASS CLASH is shown in period 1
Monday. In this period class 1 appears more than once. There is no scheduling problem between
Periods 1 and 2 because they are different time slots.
It is possible to define an objective or cost function for evaluating a given timetable. This
function is an arbitrary measure of the quality of the solution. A convenient cost function
calculates the number of clashes in any given timetable. An acceptable timetable has a cost of 0.
The optimisation problem becomes one of minimising the value of this cost function. The cost of
any period can be expressed as the sum of three components corresponding to a class cost, a
teacher cost and a room cost. It is not strictly necessary to sum the components, providing they
can be combined to reflect the quality of the solution. However, by using a sum, it then becomes
easy to weight the various costs so that one may be made more important that the others. In this
way the optimisation process can be guided towards a solution in which some types of clash are
more important than others. This weighting technique is discussed in section 5.5.
The class cost is the number of times each of the classes in the period appears in that
period, less one if it is greater than zero. Thus, if a class appears no times or once in a period
then the cost of that class is zero. If it appears many times the the class cost for that class is the
number of times less one. The class cost for a period is the sum of all class costs. The same
Solving Timetables using Simulated Annealing Page 3
computation applies for teachers and rooms. The cost of the total timetable is the sum of the
period costs.
CLASS
CLASH
Class 1,Rm9
Teacher 2
NO
CLASH
Monday Tuesday Wednesday Thursday Friday
Class 1,Rm 5
Teacher 1
Period 1
Class 1,Rm 3
Teacher 2
Period 2
Period 3
Period 4
Period 5
Figure 1 - A sample timetable
3 SIMULATED ANNEALING
Simulated annealing is a Monte-Carlo technique which can be used to find solutions to
optimisation problems. A good review of the theory and practice can be found in [10]. The
technique simulates the cooling of a collection of hot vibrating atoms. When the atoms are at a
high temperature they are free to move around, and tend to move with random displacements.
However, as the mass cools the inter-particle bonds force the atoms together. When the mass is
cool, no movement is possible, and the configuration is frozen. If the mass is cooled quickly
then chance of obtaining a low cost solution is lower than if it is cooled slowly (or annealed). At
any given temperature a new configuration of atoms is accepted if the system energy is lowered.
However, if the energy is higher, then the configuration is accepted only if the probability of
such an increase is lower than that expected at the given temperature. This probability is given by
P(D E) = e-D E/KT, where K is Boltzmann's constant. These acceptance criteria are based on the
physics of annealing, which is described in [10].
Many optimisation problems can be considered as a number of objects which need to be
scheduled such that an objective function is minimised. The vibrating atoms are replaced by the
objects, and the value of the objective function replaces the system energy. An initial schedule is
created by randomly scheduling the objects, and an initial cost (c0) and temperature (T0) are
computed. Subsequent permutations are created by randomly choosing a number of objects,
rearranging them, and computing a change in cost (D c). If D c £ 0 then the change is accepted.
However, if D c > 0 then the probability of that change at temperature T is calculated,
P(D c) = e -D c/T.
If the probability is greater than a randomly selected value in the range (0,1) then the
change is accepted. The most common technique for choosing a random number is to use a
pseudo-random uniformly distributed variable on the unit interval. After a number of successful
Solving Timetables using Simulated Annealing Page 4
permutations the temperature is decreased by a cooling rate, R, such that T = T * R, where 0
n n-1
£ R < 1, and T is a real number.
One of the advantages of simulated annealing over algorithms which always seek a better
solution (hill climbing algorithms) is that simulated annealing is less likely to get caught in local
minima, because the cost can increase as well as decrease.
4 APPLYING SA TO THE TIMETABLING PROBLEM
4.1 Technique
The application of simulated annealing to the timetabling problem is relatively straight
forward. The atoms are replaced by elements. The system energy is replaced by the timetable
cost. An initial allocation is made in which elements are placed in a randomly chosen period. The
initial cost and an initial temperature are computed (as described in section 4.2) . The cost is used
to reflect the quality of the timetable, just as the system energy reflects the quality of a substance
being annealed. The temperature is used to control the probability of an increase in cost and
relates to the temperature of a physical substance. At each iteration a period is chosen at random,
called the from period, and an element randomly selected from that period. Another period is
chosen at random, called the to period. The change in cost is calculated from two components:
1) The cost of removing the element from the from period
2) The cost of inserting the element in the to period.
The change in cost is the difference of these two components. The element is moved if the
change in cost is accepted, either because it lowers the system cost, or the increase is allowed at
the current temperature. Unlike the classic simulated annealing technique which would actually
swap two elements, an element is removed from one period and placed into another. This allows
the number of elements in one period to increase or decrease, and for all periods to a contain
different numbers of elements. If two elements were swapped then it would not be possible to
change the number of elements per period. If a particular schedule does require a fixed number
of elements per period, then it is possible to allocate the correct number in the initial allocation,
and then swap pairs of elements.
The cost of removing an element consists of a class cost, a teacher cost and a room cost.
Likewise, the cost of inserting an element consists of a class cost, a teacher cost and a room cost.
If after removing an element from a period the number of occurrences of that class is > 0, then
the class cost saving is 1. Similarly, if there are one or more occurrences of the teacher after that
teacher has been removed then the teacher saving is 1. This technique also applies for rooms.
The cost of inserting an element can be calculated using the same basic technique. In this way it
is possible to determine the change in cost incrementally without recalculating the cost of the
entire timetable. This attribute is particularly useful when the parallel version of the algorithm is
implemented.
Since the simulated annealing algorithm relies on a random set of permutations it cannot
guarantee that the true minimum cost value is actually found, or that two different annealing runs
will yield the same solution. It is possible to guarantee reproducible results by using a pseudo-
random number generator so that the same random sequence is generated for a given seed value.
Thus, different sequences can be generated by starting the random number generator with a
different seed value each time. It is often necessary to perform multiple runs to determine
whether the cost is the best that can be achieved. Some of the results presented in this paper were
produced by running the annealing program many times, each time with a different random
number seed. The global minimum cost solution was then chosen. The effect of the cooling rate
as well as the variance in final cost over multiple runs are addressed by the experiments in
section 6.2.
4.2 Run Time Parameters
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There are a number of parameters described in [10] which govern the simulated annealing
algorithm. The following values must be computed:
(1) Starting temperature
(2) When to decrease the temperature - maxswaps and maxsuccessswaps
(3) Termination of Run
(4) Cooling schedule - coolingrate
A technique for choosing the initial temperature is described in [11], and involves
computing the standard deviation in the changes in cost over a number of swaps. The scheme is
based on theory of physical annealing which is beyond the scope of this paper, but is described
in some detail in [11] and [10]. This technique allows the initial temperature to be high enough to
allow swaps which raise the cost to occur.
Each time the temperature is decreased a number of swaps is attempted. It is necessary to
limit the number of unsuccessful swaps at each temperature as well as the number of successful
swaps. These two limits are called maxswaps and maxsuccessswaps, and are chosen to be
proportional to the number of elements. Thus, as the problem grows, the time allowed to find a
solution also increases. It is possible that other functions may be applied to compute the values
of maxswaps and maxsuccessswaps, however this scheme seems to work quite well. At
any given temperature, the number of successful swaps is limited to maxsuccessswaps. The
total number of successful and unsuccessful swaps is limited to maxswaps .
There are two conditions for terminating a run. The first is if the cost becomes zero. In
this case an ideal timetable has been computed, and there is no point continuing. The second
condition arises if the cost has not changed for a certain number of iterations. The program
implemented allows an arbitrary number of temperature changes without a change in cost before
it concludes that the configuration has frozen.
The cooling schedule is the scheme used for moving from trials at one temperature to
another temperature. In the program developed a simple cooling schedule is used in which a new
temperature is computed by multiplying a coolingrate by the current temperature. This scheme
has been found to be quite effective, and is quite widely used [10].
5 PRACTICAL CONSIDERATIONS
The are a number of practical problems associated with the simple timetable model
described in the previous section. These are:
1) It cannot handle classes which have students in common,
2) It is not flexible in assigning rooms to elements,
3) It does not allow multiple periods,
4) It does not allow one class to be scheduled always with another class.
5) It does not allow one type of clash to be more important than another.
5.1 Class Clashes
In the simple model classes are assumed not to clash with any other class. Thus, the cost
of scheduling a class to a period can be calculated by checking if that class is already scheduled
in the period (likewise for the teacher and the room). Such an assumption is impractical in real
timetables. For example, year 7 may be divided into three groups, 7A, 7B and 7C. Each group
contains different students, and thus they can be scheduled together. However, when the subject
ART is taught to year 7 it may be taught to some selected students from 7A, 7B and 7C. If we
call this class 7D, then there is no way of indicating that 7D cannot be taught simultaneously with
A, B or C.
The solution involves maintaining a class clash list for each class, and using this
information in computing the cost of a period. For example, the cost should reflect that if 7D
Solving Timetables using Simulated Annealing Page 6
appears in the same period as 7A then a clash has occurred. This clash information can be
included in the simple cost function relatively easily by adding another additive cost component
which indicates how many clashing classes have been scheduled at the same time.
5.2 Room Assignment
Static assignment of rooms to requirements removes some of the flexibility in scheduling
the timetable. For example, if a school has a number of rooms each of which hold 30 students,
then there is no reason to statically assign a particular class, subject and teacher to any one room.
It would be possible to simply choose one room from those available. This problem is solved by
using room groups. A requirement is assigned to a room group, which may contain more than
one room. When the element is scheduled to a period, the room cost does not increase until all of
the rooms in the room group have been assigned. Thus, if there are 20 rooms which can hold 30
students each, then the room cost does not increase for this room group until 21 classes are
scheduled for the group. It is not necessary to actually assign the rooms when the element is
scheduled, but simply to decrement the number available in the group. The rooms can then be
assigned once a feasible timetable has been generated.
Flexible room assignment may be performed in which a room is contained in more than
one room group. In this way, it is possible to request that classes which need special purpose
rooms receive them, but if the special purpose rooms are not required in a period then they can
be used for general subjects. The only added complication with this operation is that a
requirement which requires a room which is also in a room group needs to also claim a room
from the room group when it is scheduled.
5.3 Multiple Periods
Practical timetables often call for a class to be scheduled over more than one consecutive
time period. For example, practical subjects often need 2 or 3 hours of contiguous time. The
simple model described takes no account of multiple period assignments; a perfect timetable is
one in which there are no clashes. It is possible to include multiple periods by introducing a cost
measure which increases the timetable cost if the periods of a multiple period are not consecutive.
In this way a perfect timetable not only has no clashes, but also all multiple period assignments
are satisfied. If a new configuration wishes to split a multiple period request, then it is only
accepted if the change in cost is allowed at the given temperature.
5.4 Groups of Requirements
The complexity of the scheduling task can be greatly reduced by removing unnecessary
requirements. If it is known that a set of classes are always scheduled together in the same
period, then there is no need to have a separate requirement for each one. For example, a year
level may take a number of art/craft subjects, each of which is a separate class. However, such
elements are always scheduled together in the same period because they involve a complete set of
students. While the cost function will indicate that such requirements can be scheduled together,
there is no fixed requirement that they are. If the scheduler knows to always allocate a group of
requirements to the same period, then the entire group is moved when one is moved. The change
in cost reflects moving all of the individual elements.
5.5 Weighted Cost Function
When creating a timetable certain scheduling requirements may be more important than
others. For example, it may be more important that no class appears more than once in a period
than a room is used more than once, because it may be possible to find a spare room after the
timetable has been generated. This preference can be expressed by weighting the components of
the cost function. Thus, the cost function is the sum of the weighted cost values for class cost,
teacher cost, room cost and multiple period cost. If any one component is more important than
another then the weight can be increased appropriately.
Solving Timetables using Simulated Annealing Page 7
5.6 Preferences for periods
It may be necessary to express a preference that a requirement appear in a particular
period, or range of periods. This information can be easily added into the cost function, by
including a cost component which reflects how close a requirement is to its desired period. When
a requirement is close to its preferred period the cost is low, however, as the requirement moves
further from the period the cost is increased. A preference for more than one period may be
indicated, so that a number of preferred times can be nominated. This cost component can be
computed very quickly by looking up the preference in a statically computed cost table.
5.7 Limiting number of periods taught
Many schools wish to express a preference that a teacher is not scheduled more than a
certain number of times in a given period range. For example, a particular teacher may not wish
to teach more than 3 period on a Friday. This request can be easily incorporated into the cost
function by keeping track of how many times a teacher has been scheduled in a given period.
The cost computation requires more than one period of the teacher clash array to be consulted
when a requirement is moved.
6 SOME EXPERIMENTAL RESULTS
6.1 Test Data
In this section we show some results of the application of simulated annealing to some
randomly constructed data and some real data from an Australian High School. The simulated
annealing algorithm has been implemented with a 1300 line Pascal program. The complexity of
the code is much less than a heuristic search program which required about 4000 lines of Pascal
code [12]. The latter program did not implement many of the constraints described in section 5.
The program was tested with two types of data. One type is generated automatically by a
program which creates a random timetable with zero cost. The timetable is generated by
choosing a class, teacher and room combination from a random pool, and then placing it in a
period which does not increase the cost of the configuration. This technique means that there is
always at least one solution to the timetable, although there may be more than one legal solution.
The second type of data was taken from an existing timetable in a secondary high school.
This data was much more complex, and needed room groups, clashing classes and groups of
requirements. The results of these experiments are summarised in Table 1. Timetables 1 through
9 are random timetables with no class clashes, multiple periods or grouped requirements.
Timetables 1 through 5 do not use grouped rooms, that is, each room group only has one room
available. Timetables 6 through 9 use grouped rooms, as described in the notes below Table 1.
Timetable 10 was taken from real high school . Timetable 10 used groups of requirements,
clashing classes and room groups.
It is difficult to design a measure of timetable complexity because there are so many
variables. In general, the complexity increases as the number of elements increases, but also
decreases as the numbers of rooms, teachers and classes increase. However, grouped
requirements, room groups and class clash lists all alter the complexity of any given data set.
Consequently, a simple and effective method of computing the complexity of each of these
timetables is to calculate the initial cost. Since the initial configuration is random, the initial
cost gives quite a good measure of "how hard the timetable is to schedule". If there is a great
degree of flexibility in the possible assignments, then the initial cost is likely to be low.
However, if there is very little freedom, the initial cost will be high. The use of initial cost as a
complexity measure automatically takes into account the number of elements to be scheduled, the
number of periods available, the number of teachers, rooms and classes, and the difficulty of
scheduling classes which clash with each other. One problem with this scheme is that the initial
cost will vary for particular set of requirements depending on the random numbers chosen.
Solving Timetables using Simulated Annealing Page 8
Thus, a much more reasonable measure would be the average initial cost over a number of runs.
The initial costs shown in Tables 1 and 2 were averaged over 10 runs.
Test Number Number Number Number Average Final Cooling Execution
Data Teachers Rooms Classes Elements Initial Cost Rate Time on
Set Cost Sun 3/60
1 15 15 (a) 15 100 29.8 0 0.9 11 secs
2 15 15 (a) 15 150 63.6 0 0.9 26 secs
3 15 15 (a) 15 200 115.4 0 0.9 50 secs
4 15 15 (a) 15 250 171 0 0.9 89 secs
5 15 15 (a) 15 300 240.1 0 0.9 131 secs
6 36 24 (b) 46 400 200 0 0.9 140 secs
7 36 24 (b) 46 600 352.9 0 0.9 238 secs
8 30 24 (b) 30 600 416.7 0 0.9 296 secs
9 20 24 (b) 20 600 533.4 3 0.99 5548 secs
10 37 24 (b) 101 757 765.9 0 0.95 14 hours
Notes:
(a) indicates that all room groups contained only 1 room
(b) indicates that there were 3 rooms in one group, 14 rooms in another group and all other
groups contained one room.
All timetables contained 30 periods.
Table 1 - Results of Test Data
The results are shown in Table 1. The number of periods per week, teachers, rooms,
classes and elements is shown for each data set. The initial cost gives an indication of the
complexity of the schedule. The final cost indicates whether the program found a correct
solution. For the randomly constructed data, the final cost should be zero. The cooling rate
shows how quickly the temperature was decreased. The execution time on a Sun 3/60 computer
is shown for comparison with other scheduling techniques.
The results show that increasing the number of elements for a given number of classes,
rooms and teachers increased the complexity of the problem. In all cases more elements
produced a higher average initial cost. The numbers used in tests 1 through 5 were thought
realistic for a small school timetable, and those for tests 6 through 9 for a typical high school.
Unfortunately, the cost of Test Data 9 could only be reduced to 3, and the true minimum (0)
could not be found. However, this test data is extremely complex, and whilst an optimum value
is possible, the data has very little flexibility in the way it must be scheduled. (This is evident
because 20 classes with 30 periods has a theoretical maximum of 600 requirements for a zero
cost solution. Further, there are only 20 teachers, and 24 rooms.) Data set 10 reached its
theoretical minimum cost, but only with a very long cooling cycle. Many solutions with non zero
costs could be found with much faster cooling cycles. For example, a timetable with a few
clashes was found in a few hours using a faster cooling rate. The results shown in Table 1 were
obtained by running the program a number of times and choosing the best solution. In some
cases (Data Set 9 and 10) it was also necessary to increase the cooling rate. In section 6.2 we
illustrate the effect of the cooling rate as well as the variance obtained by multiple runs.
Solving Timetables using Simulated Annealing Page 9
Test Number Number Number Number Actual Average Final %deGans %annealing
Data Teachers Rooms Classes Elements Elements Initial Cost assigned assigned
Set Cost
1 44 28 26 860 780 734.8 1 99.4 99.8
2 75 59 72 1504 1279 1297.4 0 97.6 100.0
3 67 61 31 1463 930 711.8 0 94.2 100.0
4 58 39 27 1053 810 668.2 0 89.3 100.0
5 24 20 15 480 450 404 0 99.6 100.0
6 20 11 11 340 330 315.9 0 98.2 100.0
7 34 43 31 943 930 919.7 4 96.7 99.5
8 47 56 55 1398 1398 1390.8 5 99.2 99.6
9 20 30 25 588 588 540.3 0 99.2 100.0
10 71 48 41 1350 1230 1124.1 0 98.7 100.0
11 51 54 44 1302 1302 1281 1 99.8 99.9
12 77 58 48 1593 1440 1324.2 0 99.4 100.0
13 93 107 111 2252 2252 1910.3 0 98.3 100.0
14 44 19 28 484 484 347 0 97.3 100.0
15 71 52 33 1451 990 807.4 0 94.1 100.0
16 63 37 42 1114 1110 1037.7 0 96.6 100.0
17 65 59 91 1567 1567 1358 0 98.3 100.0
18 22 24 17 454 454 388.8 0 99.3 100.0
19 58 43 31 1029 930 804.3 0 97.4 100.0
20 65 51 29 1348 870 678.9 0 98.0 100.0
21 28 15 12 360 360 304.2 0 91.9 100.0
Note: All data was only annealed once
30 Periods per week.
Table 2 - Comparison of deGans results to simulated annealing
Table 2 contains another set of set data and results. This data was constructed using
the random timetable builder program, but the parameters were taken from a number of real
school timetables produced in [1]. The only information available was the number of elements,
teachers, classes and rooms for each data set. The actual distribution was not available. The
percentage of correctly scheduled elements for the heuristic used in [1] is shown, as well as the
percentage achieved using simulated annealing. One complication which arises from not having
the actual data is that the number of elements used in the real school data is often larger than
theoretically possible given the number of classes, teachers and rooms. For example, it is not
possible to have a cost free timetable of 860 elements as in data set 1 with only 26 classes and
30 periods, because there are only 780 unique class assignments. In practice the actual number
can be larger than the theoretical limit because some of the elements contain null classes,
teachers and rooms. These null assignments do not require an actual class, teacher or room, and
therefore do not generate any clashes. Because this information was not available, a random
timetable was generated with the theoretical maximum number of possible elements in such
cases. These are shown in a separate column of the table. The table indicates that simulated
annealing performed well on a range of school sizes, although it is not possible to make any
absolute comparisons between the heuristic in the paper and simulated annealing. It should also
be noted that each set of test data was only processed once. Thus, those solutions which did not
have zero cost may not be the lowest possible cost.
6.2 Effect of Control Parameters
Graph 1 shows a typical cooling schedule, in which the cost is plotted against the iteration
count. It shows the effect of the cooling rate on the cost at any point in time. At low temperatures
(high iteration counts) the probability of an increase in cost is too low, and the algorithm resorts
to seeking decreases in cost only. At high temperatures cost increases are allowed.
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Cost versus Iteration Number - Cooling Rate .9
400
350
300
250
Cost
200
150
100
50
0
0.0 1.0 2.0 3.0 4.0 5.0
Iteration Number x 10^6
Graph 1 - Effect of Temperature on Cost changes
A number of experiments were conducted to determine the effect of the cooling rate on the
quality of the solution. The results are shown in Table 3. In this test, Data Set 9 from Table 1
was annealed at a number of different cooling rates, and the final cost was measured. This data
was chosen because it is very hard to reach the optimal cost value, and thus a variation in result
would be expected. The data was annealed five times for each cooling rate to determine the range
of the results, and in each case the final cost is reported. The table also shows the average
number of iterations, average final cost, standard deviation of the number of iterations and the
standard deviation of the cost. The results show that increasing the cooling rate (i.e. slower
cooling) produces a better quality solution on average. They also indicate the variation in final
cost which is achieved by running the program many times, each time using a different random
number seed, as well as the variation in the number of iterations. An interesting observation is
that the standard deviation in cost is quite low for very fast cooling (0.1) but then rises for
moderate cooling (0.5) before it begins to fall again for very slow cooling (0.99). One possible
explanation is that slow cooling is effectively a greedy algorithm because only decreases in cost
are allowed. Thus the algorithm is unlikely to produce a radically different solution, and will
follow basically the same contour whilst optimising the cost function. Consequently, all greedy
paths lead to a final cost value with very little variation. However, moderate cooling does allow
cost increases in cost, and thus different runs may produce radically different solutions, each in
different parts of the search space. This yields a wide variation in final cost. At very slow
cooling rates large portions of the search space are explored, and the algorithm tends to yield a
much more consistent final solution cost. Consequently, the standard deviation is low again.
Cooling Final Cost on Run SD of Average Average # SD of
Rate 1 2 3 4 5 Cost Cost Iterations Iterations
0.1 16 19 18 16 17 1.3 17.2 740981 1160
0.5 12 9 13 20 8 4.72 12.4 1943120 412946
0.9 14 11 14 13 5 3.78 11.4 3592740 416961
0.95 10 12 10 9 6 2.19 9.4 5160820 514749
0.99 8 8 10 8 7 1.09 8.2 19932100 725045
Table 3 - Effect of cooling rate
Another test was conducted with the data from Table1, Data Set 9. In this experiment, the
cost weights were adjusted to see if they affected the quality of the final solution. The cooling
rate was fixed at 0.9. The results are shown in Table 4. Increasing the weight associated with a
particular cost value increases the importance of that cost measure, which should manifest itself